# Properties

 Label 3.18.a.b Level $3$ Weight $18$ Character orbit 3.a Self dual yes Analytic conductor $5.497$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3$$ Weight: $$k$$ $$=$$ $$18$$ Character orbit: $$[\chi]$$ $$=$$ 3.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.49666262034$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{14569})$$ Defining polynomial: $$x^{2} - x - 3642$$ x^2 - x - 3642 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 3\sqrt{14569}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta + 297) q^{2} + 6561 q^{3} + ( - 594 \beta + 88258) q^{4} + (3520 \beta + 191430) q^{5} + ( - 6561 \beta + 1948617) q^{6} + (29376 \beta + 12235784) q^{7} + ( - 133604 \beta + 65170116) q^{8} + 43046721 q^{9}+O(q^{10})$$ q + (-b + 297) * q^2 + 6561 * q^3 + (-594*b + 88258) * q^4 + (3520*b + 191430) * q^5 + (-6561*b + 1948617) * q^6 + (29376*b + 12235784) * q^7 + (-133604*b + 65170116) * q^8 + 43046721 * q^9 $$q + ( - \beta + 297) q^{2} + 6561 q^{3} + ( - 594 \beta + 88258) q^{4} + (3520 \beta + 191430) q^{5} + ( - 6561 \beta + 1948617) q^{6} + (29376 \beta + 12235784) q^{7} + ( - 133604 \beta + 65170116) q^{8} + 43046721 q^{9} + (854010 \beta - 404691210) q^{10} + ( - 128128 \beta - 493776756) q^{11} + ( - 3897234 \beta + 579060738) q^{12} + (3383424 \beta - 1259699122) q^{13} + ( - 3511112 \beta - 217782648) q^{14} + (23094720 \beta + 1255972230) q^{15} + ( - 26993736 \beta + 25305661960) q^{16} + (21313920 \beta - 17156563182) q^{17} + ( - 43046721 \beta + 12784876137) q^{18} + ( - 116754048 \beta + 40026771092) q^{19} + (196958740 \beta - 257263047540) q^{20} + (192735936 \beta + 80278978824) q^{21} + (455722740 \beta - 129851425044) q^{22} + ( - 1365533312 \beta + 148614371352) q^{23} + ( - 876575844 \beta + 427581131076) q^{24} + (1347667200 \beta + 898347630175) q^{25} + (2264576050 \beta - 817768577538) q^{26} + 282429536481 q^{27} + ( - 4675388688 \beta - 1208069610352) q^{28} + (2063000384 \beta - 235187034786) q^{29} + (5603159610 \beta - 2655179028810) q^{30} + ( - 4379766336 \beta + 1700377227296) q^{31} + ( - 15811058064 \beta + 2513249815824) q^{32} + ( - 840647808 \beta - 3239669296116) q^{33} + (23486797422 \beta - 7890201769374) q^{34} + (48693407360 \beta + 15900669077040) q^{35} + ( - 25569752274 \beta + 3799217502018) q^{36} + ( - 58058187264 \beta + 5326006090214) q^{37} + ( - 74702723348 \beta + 27196858542132) q^{38} + (22198644864 \beta - 8264885939442) q^{39} + (203822994600 \beta - 49188865789800) q^{40} + ( - 64587956096 \beta - 56688399724374) q^{41} + ( - 23036405832 \beta - 1428871953528) q^{42} + ( - 186387709824 \beta + 30818515744940) q^{43} + (281995072040 \beta - 33600387667176) q^{44} + (151524457920 \beta + 8240433801030) q^{45} + ( - 554177765016 \beta + 223188561694296) q^{46} + (61433939072 \beta - 139822820963280) q^{47} + ( - 177105901896 \beta + 166030448119560) q^{48} + (718876781568 \beta + 30234681237945) q^{49} + ( - 498090471775 \beta + 90101775230775) q^{50} + (139840629120 \beta - 112564211037102) q^{51} + (1046875513860 \beta - 374699460462052) q^{52} + ( - 343991051712 \beta - 265482019305738) q^{53} + ( - 282429536481 \beta + 83881572334857) q^{54} + ( - 1762621724160 \beta - 153660640038840) q^{55} + (279687642080 \beta + 282790173123360) q^{56} + ( - 766023308928 \beta + 262615645134612) q^{57} + (847898148834 \beta - 340353222681906) q^{58} + (2656278051328 \beta + 863762115543228) q^{59} + (1292246293140 \beta - 16\!\cdots\!40) q^{60}+ \cdots + ( - 5515490268288 \beta - 21\!\cdots\!76) q^{99}+O(q^{100})$$ q + (-b + 297) * q^2 + 6561 * q^3 + (-594*b + 88258) * q^4 + (3520*b + 191430) * q^5 + (-6561*b + 1948617) * q^6 + (29376*b + 12235784) * q^7 + (-133604*b + 65170116) * q^8 + 43046721 * q^9 + (854010*b - 404691210) * q^10 + (-128128*b - 493776756) * q^11 + (-3897234*b + 579060738) * q^12 + (3383424*b - 1259699122) * q^13 + (-3511112*b - 217782648) * q^14 + (23094720*b + 1255972230) * q^15 + (-26993736*b + 25305661960) * q^16 + (21313920*b - 17156563182) * q^17 + (-43046721*b + 12784876137) * q^18 + (-116754048*b + 40026771092) * q^19 + (196958740*b - 257263047540) * q^20 + (192735936*b + 80278978824) * q^21 + (455722740*b - 129851425044) * q^22 + (-1365533312*b + 148614371352) * q^23 + (-876575844*b + 427581131076) * q^24 + (1347667200*b + 898347630175) * q^25 + (2264576050*b - 817768577538) * q^26 + 282429536481 * q^27 + (-4675388688*b - 1208069610352) * q^28 + (2063000384*b - 235187034786) * q^29 + (5603159610*b - 2655179028810) * q^30 + (-4379766336*b + 1700377227296) * q^31 + (-15811058064*b + 2513249815824) * q^32 + (-840647808*b - 3239669296116) * q^33 + (23486797422*b - 7890201769374) * q^34 + (48693407360*b + 15900669077040) * q^35 + (-25569752274*b + 3799217502018) * q^36 + (-58058187264*b + 5326006090214) * q^37 + (-74702723348*b + 27196858542132) * q^38 + (22198644864*b - 8264885939442) * q^39 + (203822994600*b - 49188865789800) * q^40 + (-64587956096*b - 56688399724374) * q^41 + (-23036405832*b - 1428871953528) * q^42 + (-186387709824*b + 30818515744940) * q^43 + (281995072040*b - 33600387667176) * q^44 + (151524457920*b + 8240433801030) * q^45 + (-554177765016*b + 223188561694296) * q^46 + (61433939072*b - 139822820963280) * q^47 + (-177105901896*b + 166030448119560) * q^48 + (718876781568*b + 30234681237945) * q^49 + (-498090471775*b + 90101775230775) * q^50 + (139840629120*b - 112564211037102) * q^51 + (1046875513860*b - 374699460462052) * q^52 + (-343991051712*b - 265482019305738) * q^53 + (-282429536481*b + 83881572334857) * q^54 + (-1762621724160*b - 153660640038840) * q^55 + (279687642080*b + 282790173123360) * q^56 + (-766023308928*b + 262615645134612) * q^57 + (847898148834*b - 340353222681906) * q^58 + (2656278051328*b + 863762115543228) * q^59 + (1292246293140*b - 1687902854909940) * q^60 + (50768356608*b + 1392143828013950) * q^61 + (-3001167829088*b + 1079291378249568) * q^62 + (1264540476096*b + 526710380064264) * q^63 + (-3671011095840*b - 497266784711648) * q^64 + (-3786452053120*b + 1320461339905620) * q^65 + (2989996897140*b - 851955199713684) * q^66 + (534402316800*b - 1664650838348092) * q^67 + (12072122481468*b - 3174257240883036) * q^68 + (-8959264060032*b + 975058890440472) * q^69 + (-1438727091120*b - 1662229550569680) * q^70 + (-1573308147840*b - 6744701103252408) * q^71 + (-5751214112484*b + 2805359800989636) * q^72 + (36563049105408*b + 218294959068362) * q^73 + (-22569287707622*b + 9194471381036502) * q^74 + (8842044499200*b + 5894058801578175) * q^75 + (-34080380797032*b + 12626173834555688) * q^76 + (-16072932516608*b - 6535270505868192) * q^77 + (14857883464050*b - 5365379637226818) * q^78 + (-29178746268480*b + 2188020782607440) * q^79 + (83908519216720*b - 7614585847354320) * q^80 + 1853020188851841 * q^81 + (37505776763862*b - 8367617326875462) * q^82 + (-15615881792128*b - 19943221132806444) * q^83 + (-30675225181968*b - 7926144713519472) * q^84 + (-56310978695040*b + 6553071925276140) * q^85 + (-86175665562668*b + 33592442076079884) * q^86 + (13535345519424*b - 1543062135230946) * q^87 + (57620433085776*b - 29934904994740944) * q^88 + (113966194036992*b + 3486092048053722) * q^89 + (36762330201210*b - 17420629608022410) * q^90 + (4393923836544*b - 2381098286163344) * q^91 + (-208796175633584*b + 119472162668019504) * q^92 + (-28735646930496*b + 11156174988289056) * q^93 + (158068700867664*b - 49582657351153872) * q^94 + (118544006835200*b - 46225029707742600) * q^95 + (-103736351957904*b + 16489432041621264) * q^96 + (-48656866579200*b + 91784777856230498) * q^97 + (183271722887751*b - 85280142148308063) * q^98 + (-5515490268288*b - 21255470251817076) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 594 q^{2} + 13122 q^{3} + 176516 q^{4} + 382860 q^{5} + 3897234 q^{6} + 24471568 q^{7} + 130340232 q^{8} + 86093442 q^{9}+O(q^{10})$$ 2 * q + 594 * q^2 + 13122 * q^3 + 176516 * q^4 + 382860 * q^5 + 3897234 * q^6 + 24471568 * q^7 + 130340232 * q^8 + 86093442 * q^9 $$2 q + 594 q^{2} + 13122 q^{3} + 176516 q^{4} + 382860 q^{5} + 3897234 q^{6} + 24471568 q^{7} + 130340232 q^{8} + 86093442 q^{9} - 809382420 q^{10} - 987553512 q^{11} + 1158121476 q^{12} - 2519398244 q^{13} - 435565296 q^{14} + 2511944460 q^{15} + 50611323920 q^{16} - 34313126364 q^{17} + 25569752274 q^{18} + 80053542184 q^{19} - 514526095080 q^{20} + 160557957648 q^{21} - 259702850088 q^{22} + 297228742704 q^{23} + 855162262152 q^{24} + 1796695260350 q^{25} - 1635537155076 q^{26} + 564859072962 q^{27} - 2416139220704 q^{28} - 470374069572 q^{29} - 5310358057620 q^{30} + 3400754454592 q^{31} + 5026499631648 q^{32} - 6479338592232 q^{33} - 15780403538748 q^{34} + 31801338154080 q^{35} + 7598435004036 q^{36} + 10652012180428 q^{37} + 54393717084264 q^{38} - 16529771878884 q^{39} - 98377731579600 q^{40} - 113376799448748 q^{41} - 2857743907056 q^{42} + 61637031489880 q^{43} - 67200775334352 q^{44} + 16480867602060 q^{45} + 446377123388592 q^{46} - 279645641926560 q^{47} + 332060896239120 q^{48} + 60469362475890 q^{49} + 180203550461550 q^{50} - 225128422074204 q^{51} - 749398920924104 q^{52} - 530964038611476 q^{53} + 167763144669714 q^{54} - 307321280077680 q^{55} + 565580346246720 q^{56} + 525231290269224 q^{57} - 680706445363812 q^{58} + 17\!\cdots\!56 q^{59}+ \cdots - 42\!\cdots\!52 q^{99}+O(q^{100})$$ 2 * q + 594 * q^2 + 13122 * q^3 + 176516 * q^4 + 382860 * q^5 + 3897234 * q^6 + 24471568 * q^7 + 130340232 * q^8 + 86093442 * q^9 - 809382420 * q^10 - 987553512 * q^11 + 1158121476 * q^12 - 2519398244 * q^13 - 435565296 * q^14 + 2511944460 * q^15 + 50611323920 * q^16 - 34313126364 * q^17 + 25569752274 * q^18 + 80053542184 * q^19 - 514526095080 * q^20 + 160557957648 * q^21 - 259702850088 * q^22 + 297228742704 * q^23 + 855162262152 * q^24 + 1796695260350 * q^25 - 1635537155076 * q^26 + 564859072962 * q^27 - 2416139220704 * q^28 - 470374069572 * q^29 - 5310358057620 * q^30 + 3400754454592 * q^31 + 5026499631648 * q^32 - 6479338592232 * q^33 - 15780403538748 * q^34 + 31801338154080 * q^35 + 7598435004036 * q^36 + 10652012180428 * q^37 + 54393717084264 * q^38 - 16529771878884 * q^39 - 98377731579600 * q^40 - 113376799448748 * q^41 - 2857743907056 * q^42 + 61637031489880 * q^43 - 67200775334352 * q^44 + 16480867602060 * q^45 + 446377123388592 * q^46 - 279645641926560 * q^47 + 332060896239120 * q^48 + 60469362475890 * q^49 + 180203550461550 * q^50 - 225128422074204 * q^51 - 749398920924104 * q^52 - 530964038611476 * q^53 + 167763144669714 * q^54 - 307321280077680 * q^55 + 565580346246720 * q^56 + 525231290269224 * q^57 - 680706445363812 * q^58 + 1727524231086456 * q^59 - 3375805709819880 * q^60 + 2784287656027900 * q^61 + 2158582756499136 * q^62 + 1053420760128528 * q^63 - 994533569423296 * q^64 + 2640922679811240 * q^65 - 1703910399427368 * q^66 - 3329301676696184 * q^67 - 6348514481766072 * q^68 + 1950117780880944 * q^69 - 3324459101139360 * q^70 - 13489402206504816 * q^71 + 5610719601979272 * q^72 + 436589918136724 * q^73 + 18388942762073004 * q^74 + 11788117603156350 * q^75 + 25252347669111376 * q^76 - 13070541011736384 * q^77 - 10730759274453636 * q^78 + 4376041565214880 * q^79 - 15229171694708640 * q^80 + 3706040377703682 * q^81 - 16735234653750924 * q^82 - 39886442265612888 * q^83 - 15852289427038944 * q^84 + 13106143850552280 * q^85 + 67184884152159768 * q^86 - 3086124270461892 * q^87 - 59869809989481888 * q^88 + 6972184096107444 * q^89 - 34841259216044820 * q^90 - 4762196572326688 * q^91 + 238944325336039008 * q^92 + 22312349976578112 * q^93 - 99165314702307744 * q^94 - 92450059415485200 * q^95 + 32978864083242528 * q^96 + 183569555712460996 * q^97 - 170560284296616126 * q^98 - 42510940503634152 * q^99

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 60.8511 −59.8511
−65.1063 6561.00 −126833. 1.46604e6 −427163. 2.28730e7 1.67913e7 4.30467e7 −9.54488e7
1.2 659.106 6561.00 303349. −1.08318e6 4.32440e6 1.59855e6 1.13549e8 4.30467e7 −7.13934e8
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.18.a.b 2
3.b odd 2 1 9.18.a.c 2
4.b odd 2 1 48.18.a.h 2
5.b even 2 1 75.18.a.b 2
5.c odd 4 2 75.18.b.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.18.a.b 2 1.a even 1 1 trivial
9.18.a.c 2 3.b odd 2 1
48.18.a.h 2 4.b odd 2 1
75.18.a.b 2 5.b even 2 1
75.18.b.c 4 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 594T_{2} - 42912$$ acting on $$S_{18}^{\mathrm{new}}(\Gamma_0(3))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 594T - 42912$$
$3$ $$(T - 6561)^{2}$$
$5$ $$T^{2} - 382860 T - 1587996193500$$
$7$ $$T^{2} - 24471568 T + 36563624964160$$
$11$ $$T^{2} + 987553512 T + 24\!\cdots\!72$$
$13$ $$T^{2} + 2519398244 T + 85\!\cdots\!88$$
$17$ $$T^{2} + 34313126364 T + 23\!\cdots\!24$$
$19$ $$T^{2} - 80053542184 T - 18\!\cdots\!20$$
$23$ $$T^{2} - 297228742704 T - 22\!\cdots\!20$$
$29$ $$T^{2} + 470374069572 T - 50\!\cdots\!80$$
$31$ $$T^{2} - 3400754454592 T + 37\!\cdots\!00$$
$37$ $$T^{2} - 10652012180428 T - 41\!\cdots\!20$$
$41$ $$T^{2} + 113376799448748 T + 26\!\cdots\!40$$
$43$ $$T^{2} - 61637031489880 T - 36\!\cdots\!96$$
$47$ $$T^{2} + 279645641926560 T + 19\!\cdots\!36$$
$53$ $$T^{2} + 530964038611476 T + 54\!\cdots\!20$$
$59$ $$T^{2} + \cdots - 17\!\cdots\!80$$
$61$ $$T^{2} + \cdots + 19\!\cdots\!56$$
$67$ $$T^{2} + \cdots + 27\!\cdots\!64$$
$71$ $$T^{2} + \cdots + 45\!\cdots\!64$$
$73$ $$T^{2} - 436589918136724 T - 17\!\cdots\!00$$
$79$ $$T^{2} + \cdots - 10\!\cdots\!00$$
$83$ $$T^{2} + \cdots + 36\!\cdots\!72$$
$89$ $$T^{2} + \cdots - 16\!\cdots\!60$$
$97$ $$T^{2} + \cdots + 81\!\cdots\!04$$